A Numerical Study of Uniform Superconvergence of LDG Method for Solving Singularly Perturbed Problems (Journal of Computational Mathematics, 2018)
In this paper, we consider the local discontinuous Galerkin method (LDG) for solvingsingularly perturbed convection-diffusion problems in one- and two-dimensional settings.The existence and uniqueness of the LDG solutions are verified. Numerical experimentsdemonstrate that it seems impossible to obtain uniform superconvergence fornumerical fluxes under uniform meshes. Thanks to the implementation of two-type differentanisotropic meshes, i.e., the Shishkin and an improved grade meshes, the uniform $2p + 1$-order superconvergence is observed numerically for both one-dimensional and two-dimensionalcases.
Keywords: Singularly perturbed problems; Local discontinuous Galerkin method; Numerical fluxes; Uniform superconvergence.